Astrophysics (Index) | About |

Ludwig Eduard Boltzmann is responsible for a lot of equations
and in various fields, different equations are termed the
**Boltzmann equation** (see below).
In the field of stellar structure, the term
(or **Boltzmann relation**, which is also ambiguous.)
is used for an equation giving
the ratio of counts of atoms at different possible atomic excitation
levels. One form:

N^{i+1}g^{i+1}———— = ———— e^{-(Ei+1-Ei)/kT}N^{i}g^{i}

- N
^{i+1}, N^{i}- number density of two states of excitation i and i+1. - g
^{i+1}, g^{i}- quantum-mechanical degeneracy weights. - k - Boltzmann constant.
- T - temperature.

This equation is related to the **Boltzmann distribution**
(frequency of particles at various states,
*not* the same as the Maxwell-Boltzmann distribution,
which is the distribution of (say) the kinetic energy
of particles in (say) a gas) incorporating
a **Boltzmann factor** (e^{-(E1-E2)/kT})
giving the relative probability of two energy levels,
and relating the ratio of the two
terms, N_{x}/g_{x}
(the number of atoms in specific quantum state)
to it.
This *Boltzmann equation* is analogous to the Saha equation
which gives similar ratios for states of ionization.

Another **Boltzmann equation** used in astrophysics is
the **Boltzmann Transport Equation** (**BTE**, often shortened
to *Boltzmann equation*), which is more general than the
Maxwell-Boltzmann distribution, handling gases
not necessarily in thermodynamic equilibrium.
(It seems plausible to me that the above relation is derived
from this, a possible reason for the "name clash".)

http://en.wikipedia.org/wiki/Boltzmann_equation

http://spiff.rit.edu/classes/phys440/lectures/boltz/boltz.html

Boltzmann Transport Equation (BTE)

degeneracy weight

state of excitation

Stefan-Boltzmann constant (σ)

stellar dynamics

temperature